0,3

5^a(n) is the highest power of 5 dividing (5^n)! - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 04 2002

n such that A002294(n) is not divisible by 5 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 14 2003

Numbers n such that a(n) is prime are listed in A004061(n) = {3,7,11,13,47,127,149,181,619,929,...}. Corresponding prime a(n) are listed in A086122(n) = {31,19531,12207031,305175781,177635683940025046467781066894531,...}. 3^(m+1) divides a(2*3^m*k). 31 divides a(3k). 13 divides a(4k). 11 divides a(5k). 71 divides a(5k). 7 divides a(6k). 19531 divides a(7k). 313 divides a(8k). 19 divides a(9k). 829 divides a(9k). 71 divides a(10k). 521 divides a(10k). 17 divides a(16k). p divides a(p-1) for all prime p except p = {2,5}. p^(m+1) divides a(p^m*(p-1)) for all prime p except p = {2,5}. p divides a((p-1)/2) for prime p = {11,19,29,31,41,59,61,71,79,89,101,109,...} = A045468 Primes congruent to {1, 4} mod 5. p divides a((p-1)/3) for prime p = {13,67,127,163,181,199,211,241,313,337,367,379,409,457,...}. p divides a((p-1)/4) for prime p = {101,109,149,181,269,389,401,409,449,461,521,541,...} = A107219 Primes of the form x^2+100y^2. p divides a((p-1)/5) for prime p = {31,191,251,271,601,641,761,1091,1861,...}. p divides a((p-1)/6) for prime p = {181,199,211,241,379,409,631,691,739,769,1039,...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 23 2007

Starting with 1 = convolution square of A026375: (1, 3, 11, 45, 195, 873,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]

Except for the first term, a(n)=5*a(n-1)+1 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 29 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 374

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Second binomial transform of A015518; binomial transform of A000302 (preceded by 0). - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003

a(n)=sum{k=1..n, C(n, k)4^(k-1) } - Paul Barry (pbarry(AT)wit.ie), Mar 28 2003

Without leading zero, this is (5*5^n-1)/4, the binomial transform of A003947. - Paul Barry (pbarry(AT)wit.ie), May 19 2003

a(n) = (-1)^n times the (i, j)-th element of M^n (for all i and j such that i is not equal to j), where M = ((1, -1, 1, -2), (-1, 1, -2, 1), (1, -2, 1, -1), (-2, 1, -1, 1)). - Simone Severini (ss54(AT)york.ac.uk), Nov 25 2004

a(n) = A125118(n,4) for n>3. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 21 2006

((3+sqrt4)^n-(3-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=31. [From Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008]

a(n)=6*a(n-1)-5*a(n-2), n>1 ; a(0)=0, a(1)=1 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 01 2009]

Base 5...........decimal (comment from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 14 2007):

0......................0

1......................1

11.....................6

111...................31

1111.................156

11111................781

111111..............3906

1111111............19531

11111111...........97656

111111111.........488281

1111111111.......2441406

etc. ...............etc.

a:=n->sum(5^(n-j), j=1..n): seq(a(n), n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 04 2007

A003463:=1/(5*z-1)/(z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=6*a[n-1]-5*a[n-2]od: seq(a[n], n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 21 2008

lst={}; Do[p=(5^n-1)/4; AppendTo[lst, p], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 29 2008]

(Other) sage: [lucas_number1(n, 6, 5) for n in xrange(0, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

(Other) sage: [gaussian_binomial(n, 1, 5) for n in xrange(0, 24)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 28 2009]

Cf. A004061, A086122, A045468, A107219, A074479.

A026375 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009]

Sequence in context: A056015 A128740 A026705 this_sequence A026771 A065096 A077352

Adjacent sequences: A003460 A003461 A003462 this_sequence A003464 A003465 A003466

nonn

N. J. A. Sloane (njas(AT)research.att.com).